Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

Abstract

We wish to use graded structures [KrVu87], [Vu01] on differential operators and quasimodular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corresponding non-archimedean weight spaces.An approach to constructions of automorphic L-functions on unitary groups and their padic avatars is presented. For an algebraic group G over a number field K these L functions are certain Euler products L(s, π, r, χ). In particular, our constructions cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro and Rallis.A p-adic analogue of L(s, π, r, χ) is a p-adic analytic function Lp(s, π, r, χ) of p-adic arguments s ∈ Zp, χ mod p r which interpolates algebraic numbers defined through the normalized critical values L * (s, π, r, χ) of the corresponding complex analytic Lfunction. We present a method using arithmetic nearly-holomorphic forms and general quasimodular forms, related to algebraic automorphic forms. It gives a technique of constructing p-adic zeta-functions via quasi-modular forms and their Fourier coefficients.